Problem: The equation of hyperbola $H$ is $\dfrac {(x-1)^{2}}{9}-\dfrac {(y-2)^{2}}{4} = 1$. What are the asymptotes?
Solution: We want to rewrite the equation in terms of $y$ , so start off by moving the $y$ terms to one side: $\dfrac {(y-2)^{2}}{4} = - 1 + \dfrac {(x-1)^{2}}{9}$ Multiply both sides of the equation by $4$ $(y-2)^{2} = { - 4 + \dfrac{ (x-1)^{2} \cdot 4 }{9}}$ Take the square root of both sides. $\sqrt{(y-2)^{2}} = \pm \sqrt { - 4 + \dfrac{ (x-1)^{2} \cdot 4 }{9}}$ $ y - 2 = \pm \sqrt { - 4 + \dfrac{ (x-1)^{2} \cdot 4 }{9}}$ As $x$ approaches positive or negative infinity, the constant term in the square root matters less and less, so we can just ignore it. $y - 2 \approx \pm \sqrt {\dfrac{ (x-1)^{2} \cdot 4 }{9}}$ $y - 2 \approx \pm \left(\dfrac{2 \cdot (x - 1)}{3}\right)$ Add $2$ to both sides and rewrite as an equality in terms of $y$ to get the equation of the asymptotes: $y = \pm \dfrac{2}{3}(x - 1)+ 2$